STA Module 8 The Sampling Distribution of the Sample Mean. Rev.F08 1

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1 STA 2023 Module 8 The Sampling Distribution of the Sample Mean Rev.F08 1

2 Module Objectives 1. Define sampling error and explain the need for sampling distributions. 2. Find the mean and standard deviation of the variable, given the mean and standard deviation of the population and the sample size. 3. State and apply the central limit theorem. 4. Determine the sampling distribution of the sample mean when the variable under consideration is normally distributed. 5. Determine the sampling distribution of the sample mean when the sample size is relatively large. Rev.F08 2

3 Why Sampling? We have seen that using a sample to acquire information about a population is often preferable to conducting a census. Generally, sampling is less costly and can be done more quickly than a census; it is often the practical way to gather information. Rev.F08 3

4 What is Sampling Error? However, because a sample provides data for only a portion of an entire population, we cannot expect the sample to yield perfectly accurate information about the population. Thus, we should anticipate that a certain amount of error - called sampling error - will result simply because we are sampling. Rev.F08 4

5 What is a Sampling Distribution? The distribution of all possible observations of the statistic for samples of a given size is called the sampling distribution of the statistic. For example, the sampling distribution of the sample mean is the distribution of all possible sample means for samples of a given size. In this module, we will concentrate on the sampling distribution of the sample mean. Rev.F08 5

6 Sampling Distribution of the Sample Mean What does it mean? The sampling distribution of the sample mean is the distribution of all possible sample means for samples of a given size. Rev.F08 6

7 Example: Sample Means for Samples of Size 2 Rev.F08 7

8 Dotplot for the Sampling Distribution This is a dotplot for the sampling distribution of the sample mean for samples of size 2 in the previous example. The dotplot shows that 3 out of 10 samples have means within 1 inch of the population mean of 80 inches. Rev.F08 8

9 Dotplot for the Sampling Distribution of the Sample Mean As we can see here, the possible sample means cluster more closely around the population mean as the sample size increases. This suggests that sampling error tends to be smaller for large samples than for small samples. Rev.F08 9

10 Sample Size and Sampling Error As we can see here, the larger the sample size (first column), the percentage of sample means lie within half an inch from the population mean (last column) is getting larger. This means that the larger the sample size, the smaller the sampling error. Rev.F08 10

11 What is the Mean of the Sample Mean? In short, the mean of all possible sample means equal to the population mean. Rev.F08 11

12 What is the Standard Deviation of the Sample Mean? In short, for each sample size, the standard deviation of all possible sample means equals to the population standard deviation divided by the square root of the sample size. Rev.F08 12

13 Sampling Distribution of the Sample Mean for a Normally distributed Variable The possible sample mean IQs for samples of four people have a normal distribution with mean 100 and standard deviation 8, whereas the possible sample mean IQs for samples of 16 people have a normal distribution with mean 100 and standard deviation 4. Thus, the larger the sample size, the smaller the sampling error tends to be in estimating a population mean by sample mean. Rev.F08 13

14 Simulating the Sampling Distribution of a Mean Let s look at the sampling distribution of the sample mean again. Like any statistic computed from a random sample, a sample mean also has a sampling distribution. We can use simulation to get a sense as to what the sampling distribution of the sample mean might look like Rev.F08 14

15 Mean - The Average of One Die Now, let s go through a simulation of 10,000 tosses of a die. A histogram of the results is: Rev.F08 15

16 Means Averaging Two Dice Looking at the average of two dice after a simulation of 10,000 tosses: Rev.F08 16

17 Means Averaging Three Dice The average of three dice after a simulation of 10,000 tosses looks like: Rev.F08 17

18 Means Averaging Five Dice The average of 5 dice after a simulation of 10,000 tosses looks like: Rev.F08 18

19 Means Averaging Twenty Dice The average of 20 dice after a simulation of 10,000 tosses looks like: Rev.F08 19

20 What the Simulations Show? As the sample size (number of dice) gets larger, each sample average (sample mean) is more likely to be closer to the population mean. So, we see the shape continuing to tighten around 3.5 And, it probably does not shock you that the sampling distribution of a mean becomes Normal. Rev.F08 20

21 The Central Limit Theorem: The Fundamental Theorem of Statistics The sampling distribution of any mean becomes more nearly Normal as the sample size grows. All we need is for the observations to be independent and collected with randomization. We don t even care about the shape of the population distribution! The Fundamental Theorem of Statistics is called the Central Limit Theorem (CLT). Rev.F08 21

22 The Central Limit Theorem (cont.) The CLT is surprising and a bit weird: Not only does the histogram of the sample means get closer and closer to the Normal model as the sample size grows, but this is true regardless of the shape of the population distribution. The CLT works better (and faster) the closer the population model is to a Normal itself. It also works better for larger samples. Rev.F08 22

23 The Central Limit Theorem (CLT) The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better the approximation will be. Rev.F08 23

24 The Real World and the Model World Now we have two distributions to deal with. The first is the real world distribution of the sample, which we might display with a histogram. The second is the math world sampling distribution of the statistic, which we model with a Normal model based on the Central Limit Theorem. Don t confuse the two! Rev.F08 24

25 Normal Model The CLT says that the sampling distribution of any mean is approximately Normal. It s centered at the population mean. But what about the standard deviations? Rev.F08 25

26 Normal Model (cont.) The Normal model for the sampling distribution of the mean has a standard deviation equal to! SD( y) = n where σ is the population standard deviation. Rev.F08 26

27 Standard Deviation and Standard Error When we don t know the population standard deviation σ, are we stuck? Nope! We can use sample statistics to estimate these population parameters. Whenever we estimate the standard deviation of a sampling distribution, we call it a standard error. Rev.F08 27

28 Sampling Distribution Models Always remember that the statistic itself is a random quantity. We can t know what our statistic will be because it comes from a random sample. Fortunately, for the mean, the CLT tells us that we can model their sampling distribution directly with a Normal model. Rev.F08 28

29 Sampling Distribution Models (cont.) There are two basic truths about sampling distributions: 1. Sampling distributions arise because samples vary. Each random sample will have different cases and so, a different value of the statistic. 2. Although we can always simulate a sampling distribution, the Central Limit Theorem saves us the trouble for means and proportions (we will look at proportions later.) Rev.F08 29

30 The Process Going Into the Sampling Distribution Model Rev.F08 30

31 Important Key Fact Rev.F08 31

32 What Can Go Wrong? Don t confuse the sampling distribution with the distribution of the sample. When you take a sample, you look at the distribution of the values, usually with a histogram, and you may calculate summary statistics. The sampling distribution is an imaginary collection of the values that a statistic might have taken for all random samples the one you got and the ones you didn t get. Rev.F08 32

33 What Can Go Wrong? (cont.) Beware of observations that are not independent. The CLT depends crucially on the assumption of independence. You can t check this with your data you have to think about how the data were gathered. Watch out for small samples from skewed populations. The more skewed the distribution, the larger the sample size we need for the CLT to work. Rev.F08 33

34 We have learned to: What have we learned? 1. Define sampling error and explain the need for sampling distributions. 2. Find the mean and standard deviation of the variable, given the mean and standard deviation of the population and the sample size. 3. State and apply the central limit theorem. 4. Determine the sampling distribution of the sample mean when the variable under consideration is normally distributed. 5. Determine the sampling distribution of the sample mean when the sample size is relatively large. Rev.F08 34

35 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbooks. Weiss, Neil A., Introductory Statistics, 8th Edition Weiss, Neil A., Introductory Statistics, 7th Edition Bock, David E., Stats: Data and Models, 2nd Edition Rev.F08 35

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